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Calculate the limit: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4x + 1) (2023)
Calculate the limit: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4x + 1) (2023)
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Q1
Calculate the limit: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4x + 1) (2023)
3/5
5/3
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Dividing numerator and denominator by x^2 gives lim (x -> ∞) (3 + 2/x^2)/(5 - 4/x + 1/x^2) = 3/5.
Questions & Step-by-step Solutions
1 item
Q
Q: Calculate the limit: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4x + 1) (2023)
Solution:
Dividing numerator and denominator by x^2 gives lim (x -> ∞) (3 + 2/x^2)/(5 - 4/x + 1/x^2) = 3/5.
Steps: 8
Show Steps
Step 1: Identify the limit we want to calculate: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4x + 1).
Step 2: Notice that both the numerator (3x^2 + 2) and the denominator (5x^2 - 4x + 1) are polynomials of degree 2.
Step 3: To simplify the limit, divide every term in the numerator and the denominator by x^2, the highest power of x in the expression.
Step 4: Rewrite the expression: (3x^2/x^2 + 2/x^2) / (5x^2/x^2 - 4x/x^2 + 1/x^2).
Step 5: This simplifies to (3 + 2/x^2) / (5 - 4/x + 1/x^2).
Step 6: Now, take the limit as x approaches infinity. As x becomes very large, 2/x^2 approaches 0, -4/x approaches 0, and 1/x^2 approaches 0.
Step 7: Therefore, the limit simplifies to (3 + 0) / (5 + 0) = 3/5.
Step 8: Conclude that the limit is 3/5.
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